Math 428 - Graphs and Graph Models
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Questions and Answers

What is the sum of degrees of all vertices of a graph?

The sum of degrees of all vertices in a graph is twice the number of edges. This is known as the Handshaking Lemma.

What is the definition of a directed graph?

A directed graph is a graph where every edge has a direction associated with it. This means that each edge has a starting vertex (tail) and an ending vertex (head), unlike undirected graphs where edges connect vertices without a specified direction.

Which of the following is NOT a valid type of walk in a graph?

  • Circuit
  • Trail
  • Loop (correct)
  • Path

    • Which of these descriptions correctly defines a cycle in a graph?

    <p>A circuit that repeats no vertex except for the first and last (A)</p> Signup and view all the answers

    Which of the following statements BEST describes the relationship between a subgraph and an induced subgraph?

    <p>All induced subgraphs are subgraphs (C)</p> Signup and view all the answers

    What is the definition of the diameter of a connected graph?

    <p>The diameter of a connected graph is the greatest distance between any two vertices in the graph. This distance is determined by the shortest path connecting those two vertices.</p> Signup and view all the answers

    What is the definition of a geodesic in a graph?

    <p>A geodesic in a graph is a path between two vertices that has the shortest possible length. This shortest length is equivalent to the distance between those two vertices.</p> Signup and view all the answers

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    Flashcards

    Graph

    A graph G = (V, E) consists of vertices V and edges E.

    Vertices

    The finite set of points in a graph, represented as V.

    Edges

    The finite set of connections between vertices in a graph, represented as E.

    Order of a Graph

    The number of vertices in a graph, denoted as n.

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    Size of a Graph

    The number of edges in a graph, denoted as m.

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    Adjacent Vertices

    Two vertices are adjacent if they are connected by an edge.

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    Incident Edges

    An edge is incident to vertices if it connects them.

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    Degree of a Vertex

    The degree of a vertex u is the number of edges attached to it.

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    Sum of Degrees

    The total of all vertex degrees in a graph.

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    Directed Graph

    A graph where each edge has a specified direction.

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    Undirected Graph

    A graph where edges have no specific direction.

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    Walk

    A sequence of vertices where consecutive vertices are adjacent.

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    Trail

    A walk in which no edges are repeated.

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    Path

    A walk where all vertices are distinct (no repeats).

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    Circuit

    A closed trail of length 3 or more.

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    Cycle

    A circuit that returns to the starting vertex without repeating others.

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    Odd Cycle

    A cycle with an odd length (e.g. 3, 5).

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    Even Cycle

    A cycle with an even length (e.g. 4, 6).

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    Connected Graph

    A graph where every pair of vertices is connected by a path.

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    Subgraph

    A graph formed from a subset of vertices and edges of another graph.

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    Proper Subgraph

    A subgraph that has at least one fewer vertex and edge than the original.

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    Induced Subgraph

    A subgraph including specific vertices and all edges connecting them in the original graph.

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    Geodesic

    A path between two vertices that is the shortest possible path.

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    Distance Between Vertices

    The smallest length of any path connecting two vertices.

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    Diameter of Graph

    The greatest distance between any two vertices in a connected graph.

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    Theorem 1.6

    If a graph contains a u-v walk of length l, there is a u-v path of length at most l.

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    Circuit Properties

    Closed trails that provide unique paths back to a starting point.

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    Adjacent Vertices Properties

    Connected by an edge with no separation; defines neighborhood in a graph.

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    Study Notes

    Math 428 - Graphs and Graph Models

    • Graphs are represented as ordered pairs G = (V, E). V is a finite, non-empty set of vertices, and E is a finite set of edges. Edges are defined by connected pairs of vertices.
    • The order (n) of a graph is the number of vertices. The size (m) is the number of edges.
    • Vertices are adjacent if they are connected by an edge.
    • An edge (a,b) is incident to vertices a and b.
    • The degree of vertex u (deg(u)) is the number of vertices adjacent to u.

    Handshaking Lemma

    • For any graph, the sum of the degrees of all vertices is equal to twice the number of edges. Mathematically, ∑ deg(u) = 2m, where the sum is taken over all vertices in the graph.

    Types of Graphs

    • Directed graph: Edges have a direction.
    • Undirected graph: Edges do not have a direction.

    Walks, Trails, Paths, Circuits, and Cycles

    • A walk is a sequence of vertices where consecutive vertices are adjacent. Edges can be repeated.
    • A trail is a walk where no edges are repeated.
    • A path is a walk where no vertices are repeated.
    • A circuit is a closed trail with length 3 or more.
    • A cycle is a circuit where no vertex repeats except the first and last vertex. Odd cycles have odd length, and even cycles have even length.

    Determining Walk Type

    • Applying the definitions above, determine if given walks are paths, trails, circuits, and/or cycles.

    Subgraphs

    • A subgraph V* = (V*, E*) of a graph G = (V, E) is a graph where V* is a subset of V and E* is a subset of E.
    • A subgraph G* is a proper subgraph if V* is a strict subset of V or E* is a strict subset of E (i.e., not all vertices or edges are included).
    • An induced subgraph F of G includes all edges (u,v) of G, where u and v are vertices in F.

    Connected Graphs

    • A connected graph is a graph where every two vertices are connected by some path. A disconnected graph has more than one component.

    Distance and Diameter

    • The distance between vertices u and v is the smallest length of any path from u to v. This is denoted by d(u,v).
    • The diameter of a connected graph G (diam(G)) is the greatest distance between any two vertices in G.

    Geodesic

    • A u-v geodesic is a u-v path of length d(u,v).

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    Related Documents

    Math 428 Sections 1.1-1.2 PDF

    Description

    Explore the fundamental concepts of Graph Theory in this Math 428 quiz. Topics include the representation of graphs, the Handshaking Lemma, and the definitions of walks, trails, paths, circuits, and cycles. Test your understanding of directed and undirected graphs and their properties.

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